In recent years, with the growing concern over environmental problems the demand for good fuel economy for automobiles has been becoming stronger, and there is also a demand for excellent fuel economy for polymer compositions used for automobile tires. As a polymer composition for automobile tires, a polymer composition comprising a conjugated diene polymer such as polybutadiene or a butadiene-styrene copolymer and a filler such as carbon black, etc. is used and, for example, a polymer composition containing as the conjugated diene polymer a polymer formed by modifying with a dialkylamino group-containing acrylamide one terminal of a polymer formed by copolymerizing butadiene and styrene using an alkyllithium as a polymerization initiator (see e.g. JP-A-1-217047, U.S. Pat. No. 5,189,109) is known. Furthermore, a polymer composition containing as the conjugated diene polymer a polymer formed by modifying with bis(dimethylamino)methylvinylsilane one terminal of a polymer formed by copolymerizing butadiene and styrene using an alkyllithium as a polymerization initiator (see e.g. JP-A-1-217048, U.S. Pat. No. 5,189,109), a polymer composition containing as the conjugated diene polymer a polymer formed by modifying with a dialkylamino group-containing alkoxysilane one terminal of a polymer formed by polymerizing butadiene or copolymerizing butadiene and styrene, using an alkyllithium as a polymerization initiator (see e.g. JP-A-63-186748, U.S. Pat. No. 4,957,976 and JP-A-2005-290355, US2005-203251A1), etc. have been proposed as polymer compositions having good fuel economy.